1. Evil exists.2. Evil is a departure from the way things ought to be.3. If there is a departure from the way things ought to be, then there is a way things ought to be.4. Therefore, there is a way things ought to be.5. If there is a way things ought to be, then there is a design plan for things.6. If there is a design plan for things, then there must be a Designer.7. Therefore, there must be a Designer.

It appears that Geivett's argument is a variation on Plantinga's argument from proper function to God. In both arguments, there is a claim about the existence of normativity in the natural world that's grounded in purpose and plan. And in both arguments, there is a claim that such purpose and plan can only come from an intelligent designer (or at least that intelligent design is the only known way to get purpose and plan, and the prospects for a naturalistic account of purpose plan are unpromising).

The problem is that the claim that purpose and plan in nature requires an intelligent designer has been undercut by recent papers from Adrian Bardon, Tyler Wunder, and Peter J. Graham. These papers focus on Plantinga's use of the claim in his argument from proper function to God. But since the premise is the same in Geivett's argument, his argument from evil to God is likewise undercut.

"Suppose...that each book in the library has a number printed on its spine so as to create a one-to-one correspondence with the natural numbers. Because the collection is actually infinite, this means that every possible natural number is printed on some book. Therefore, it would be impossible to add another book to this library. For what would be the number of the new book? . . .Every possible number already has a counterpart in realty, for corresponding to every natural number is an already existent book. Therefore, there would be no number for the new book. But this is absurd, since entities that exist in reality can be numbered."[1]

We can put the argument more carefully as follows:

1. If concrete actual infinites are possible, then a library L with an infinite set of books is possible.
2. If L is possible, then it’s possible to assign a unique natural number to each book in L.
3. If it’s possible to assign a unique natural number to each book in L, then it’s possible to assign all the natural numbers to books in L without remainder.
4. If it’s possible to assign all the natural numbers to books in L without remainder, then if it’s possible to add a new book B to L, then it’s impossible to assign a unique natural number to B.
5. It’s possible to add B to L.
6. It’s possible to assign a unique natural number to B.
Therefore,
7. Concrete actual infinites are impossible.

This argument is valid.[2] Furthermore, (1), (2), (3), and (5) look to be true. Unfortunately, (4) seems false. Thus, consider library L again. Now suppose we reassign the natural numbers to the books in L as follows: assign ‘2’ to the first book, ‘3’ to the second book, and so on all the way through the rest of the books in L. Then we can free up ‘1’ to be assigned to the new book. But if so, then it is possible to assign a unique natural number to B, in which case (4) is false.[3] What does seem true, though, is not (4) but rather:

(4’) If it’s possible to assign all the natural numbers to books in L without remainder, then if it’s possible to add a new book B to L, then it’s impossible to assign a unique natural number to B if we hold fixed the original assignment of numbers to books.

Let’s revise the argument accordingly. To preserve the argument’s validity, we’ll also need to revise (6) to account for the new qualification:

(6’) It’s possible to assign a unique natural number to B (even) if we hold fixed the original assignment of numbers to books.

Does the revised version of Craig’s argument fare any better?

No, it doesn’t. For while (4’) seems clearly true, (6’) seems clearly false. For if we hold fixed the assignment of natural numbers to the books in L prior to the addition of B, then of course no unique natural number remains that can be assigned to B. But the problem here lies not with actual infinites, but rather with the internal coherence of Craig’s assertion that it must be possible to assign a unique natural number to a new book, even under the stipulation that all the unique natural numbers have already been assigned to other books.

What went wrong in Craig's argument? Recall premise (6) in the initial version of the argument:

(6) It’s possible to assign a unique natural number to B.

As Craig indicates in the quoted passage above, he accepts (6) on the grounds that:

(a) Any entity that exists in reality can be uniquely numbered.

This seems to be true. But (a) is not what Craig needs to derive (6). What he needs instead is

(b) Any entity that exists in reality can be uniquely numbered via a natural number.

Unfortunately, (b) looks to be false. For consider a library L’ that contains a set of books that can be put in a 1-1 correspondence with the irrational numbers. Such a set of books would be non-denumerably infinite; that is, it’d be actually infinite, but it couldn’t be put into a 1-1 correspondence with the natural numbers. Therefore, while all such books in L’ can be uniquely numbered, they can’t all by uniquely numbered via the natural numbers.

This example illustrates two salient points: (i) some sets of entities can’t be uniquely numbered via the natural numbers, and (ii) such entities can yet be uniquely numbered via other numbers, as there are more numbers than just the naturals. But given (i) and (ii), the door is open for numbering the new book in Craig’s library with a unique non-natural number.

We can sum up the problem with Craig's argument as follows. Either we hold fixed the assignment of natural numbers to books in Craig's infinite library or we don't. If we don't, then it's possible to reassign the natural numbers so as to free up a unique natural number for the new book. On the other hand, if we do hold fixed the original assignment of numbers to books, then it is impossible to assign a unique natural number to the new book. But of course there are more numbers than the naturals, and the new book can be numbered with one of these. If Craig yet demands that the new book be numbered with a natural number, even after all the natural numbers have been assigned to other books, then the problem lies not with the possibility of his infinite library, but rather with the coherence of the task demanded for it. Either way, then, Craig's argument is unsuccessful.

Let:
P=concrete actual infinites are possible
Q= a library L with an infinite number of books is possible
R=It’s possible to assign a unique natural number to each book in L
S=It’s possible to assign all the natural numbers to books in L without remainder
T= It’s possible to add a new book B to L
U=It’s impossible to assign a unique natural number to B

Here's the book's description from OUP:In May 2010, philosophers, family and friends gathered at the University of Notre Dame to celebrate the career and retirement of Alvin Plantinga, widely recognized as one of the world's leading figures in metaphysics, epistemology, and the philosophy of religion. Plantinga has earned particular respect within the community of Christian philosophers for the pivotal role that he played in the recent renewal and development of philosophy of religion and philosophical theology. Each of the essays in this volume engages with some particular aspect of Plantinga's views on metaphysics, epistemology, or philosophy of religion. Contributors include Michael Bergman, Ernest Sosa, Trenton Merricks, Richard Otte, Peter VanInwagen, Thomas P. Flint, Eleonore Stump, Dean Zimmerman and Nicholas Wolterstorff. The volume also includes responses to each essay by Bas van Fraassen, Stephen Wykstra, David VanderLaan, Robin Collins, Raymond VanArragon, E. J. Coffman, Thomas Crisp, and Donald Smith.

I can't believe I never posted about this, but there's an old (but excellent) review of Divine Hiddenness: New Essays at NDPR by Robert McKim.

As many of you know, McKim wrote a superb book on the problems of divine hiddenness (more specifically, the problem of religious ambiguity) and religious diversity a while back, and has another book on the problem of religious diversity forthcoming with OUP.